Module 5 Lecture - Non-parametric Comparisons for More than Two Groups

Analysis of Variance

Quinton Quagliano, M.S., C.S.P

Department of Educational Psychology

1 Overview and Introduction

Agenda

1 Overview and Introduction

2 Continuing Mean Comparisons

3 Conclusion

Agenda

1 Overview and Introduction

2 Continuing Mean Comparisons

3 Conclusion

A priori / planned / orthogonal comparisons are used to test specific contrast hypotheses established prior to collecting the data.
Contrasts are orthogonal if the tests are independent of one another: that is if the outcome of one contrast cannot predict the outcome of the other contrast. Two contrasts are orthogonal if they account for non-overlapping between group variance.
Two contrasts are orthogonal if the sum of the cross-product of the corresponding coefficients of these contrasts equals zero:
- \(\sum{(C_{j1})(C_{j2})} = 0\) where:
  - \(C_{j1}\) weight given to mean \(j\) in 1st contrast
  - \(C_{j2}\) weight given to mean \(j\) in 2nd contrast

Important: In k groups, there are only k-1 contrasts orthogonal to each other

Contrast	G1	G2	G3	G4
A	-1	-1	1	1
B	-3	1	1	1
C	-1	1	-1	1

Examples of checking cross-sums:
- Are contrast tests A and B orthogonal?:
  - \([(-1)(-3)] + [(-1)(1)] + [(1)(1)] + [(1)(1)] = 4 \rightarrow\) non-orthogonal
- Are contrast tests A and C orthogonal?:
  - \([(-1)(-1)] + [(-1)(1)] + [(1)(-1)] + [(1)(1)] = 0 \rightarrow\) orthogonal

Discuss: Are contrasts B and C orthogonal? Work through the math to show this

For the following post-hoc tests, we do not need to assess whether they are orthogonal or not

Agenda

1 Overview and Introduction

2 Continuing Mean Comparisons

3 Conclusion

Module 5 Lecture - Non-parametric Comparisons for More than Two Groups || Analysis of Variance